Explain what the archaic word "trow" means ?

The word "trow" meant something like "think" or "believe." You can find this word used in old literature fairly often. Shakespeare used the word on a few occasions. For example, see this passage from The Taming of the Shrew. (This is from the start of Act I, Scene 2.)

My best beloved and approved friend, Hortensio; and I trow this is his house. Here, sirrah Grumio, knock, I say.

What Petruchio is saying here is...

The word "trow" meant something like "think" or "believe." You can find this word used in old literature fairly often. Shakespeare used the word on a few occasions. For example, see this passage from The Taming of the Shrew. (This is from the start of Act I, Scene 2.)

My best beloved and approved friend, Hortensio; and I trow this is his house. Here, sirrah Grumio, knock, I say.

What Petruchio is saying here is that he thinks or believes that this house belongs to his friend. This would make sense if you put one of those words in there. Petruchio would be saying "I believe this is his house."

So, "trow" simply means to believe or to think that something is the case.

There are several definitions of the word trow. It was a type of cargo ship used in British rivers to transport goods, and it had a removable mast that could be taken down in order to pass under small bridges. In British folklore (particularly of the Orkney and Shetland Islands), a trow (or trowe or drow) was a mythological fairy creature, similar to a troll. Ugly and mischievous, they were believed to be mostly nocturnal creatures. It is also an ancient word that means to "think," "believe" or "suppose," or to have trust in a particular faith or belief.